We find that the photospheric radius expansion type I X-ray bursts from bright X-ray sources in globular clusters reach an empirical bolometric peak luminosity (assuming the total X-ray burst emission is entirely due to black-body radiation and the recorded maximum luminosity is the actual peak luminosity) of roughly $3.8 \times 10 ^{38}$ erg s^-1(excluding 4U 1746-37, see also below). Because significant deviations from this value occur (e.g., 4U 2129+12) this only approximately confirms an earlier suggestion by Basinska et al. (1984) that radius expansion bursts reach a "true'' critical luminosity. Our value for the average bolometric peak luminosity is very close to the "adopted standard candle'' value of $3.7 \times 10 ^{38}$ erg s^-1 by Verbunt et al. (1984) and comparable to the mean bolometric peak luminosity of $3.0 \pm 0.5 \times 10 ^{38}$ erg s^-1 by Lewin et al. (1993) of X-ray bursts with photospheric radius expansion from only four globular cluster bursters (i.e., 4U 1722-30, 4U 1820-30, A1850-08 and 4U 2129+12).

$\begin{figure} \par\includegraphics[angle=-90,width=12cm,clip]{h3715f7.ps} \end{figure}$

Figure 7: Bolometric peak luminosities reached during the brightest X-ray bursts seen in the twelve globular cluster X-ray sources, using the distances and bolometric peak fluxes as quoted in Tables 1 and 2. Photospheric radius expansion (RE) X-ray bursts have been denoted with a filled circle; the others with an open triangle. With grey bands we give the expected range in the Eddington limit for matter with cosmic composition (X=0.73) and hydrogen-poor matter (X=0), see Sect. 5.

To establish whether radius expansion bursts can be used as standard candles, one first has to verify whether the X-ray burst fluxes during every radius expansion burst in an individual source reach a single maximum value. Observations prior to RXTE and BeppoSAX revealed many (i.e., typically more than ten) radius expansion bursts in one globular cluster source, 4U 1820-30 (Vacca et al. 1986; Haberl et al. 1987; Damen et al. 1990; see Appendix B), and two sources outside globular cluster sources, MXB 1636-53 (Inoue et al. 1984; Damen et al. 1990) and MXB 1728-34 (Basinska et al. 1984). The bolometric peak fluxes were found to be the same within the errors in all three cases.

The BeppoSAX/WFCs observed many radius expansion bursts from the globular cluster sources 4U 1820-30 and 4U 1722-30. We find that the bolometric peak fluxes are also the same within the errors in both cases. A similar conclusion was reached from eight radius expansion bursts seen by the BeppoSAX/WFCs from one source outside a globular cluster, XB 1812-12 (Cocchi et al. 2000a). The more sensitive RXTE/PCA observations of especially MXB 1728-34 (e.g. van Straaten et al. 2001; Franco 2001) and MXB 1636-53 show, however, that the bolometric peak fluxes are not exactly constant, with standard deviations of $\simeq$ 9.5% and $\simeq$ 7%, respectively (Galloway et al. 2002a; 2002b). Similar studies in other sources but with less radius expansion bursts (between four to eight per source) indicate standard deviations between 7% and 15% (Muno et al. 2000; Kuulkers et al. 2002; Galloway et al. 2002a; 2002b). In the case of MXB 1728-34 it was found that the peak fluxes vary systematically on a monthly timescale; when taking this secular variation into account the residual scatter is only $\simeq$ 3% (Galloway et al. 2002a; 2002b).

For the globular cluster bursters MX 0513-40, XB 1745-25 and 4U 1746-37 only two radius expansion bursts were seen from one instrument. In all three cases similar peak fluxes within the errors were reached for both X-ray bursts (see Appendix B).

We note that two sources outside globular clusters, MXB 1659-29 (Wijnands et al. 2002) and EXO 0748-676 (Gottwald et al. 1986), showed very different peak fluxes during different radius expansion bursts. They both are viewed at a high inclination (e.g. Cominsky & Wood 1984; Parmar et al. 1986), so that, although the true critical luminosity probably is reached, the X-ray burst emission may be partly obscured by the accretion disk and/or donor star (i.e. the X-ray burst emission is anisotropic). This may lead to different observed maximum peak luminosities at different times. Note that this may also affect the peak luminosities during the radius expansion bursts from the globular cluster bursters 4U 1746-37 and GRS 1747-312, since they are also relatively high inclination systems (e.g. Sansom et al. 1993; in 't Zand et al. 2000; see also Sztajno et al. 1987, who concluded from other grounds that the X-ray burst emission from 4U 1746-37 is probably anisotropic).

Excluding the high-inclination systems, we conclude that, since multiple radius expansion bursts in a single source reach the same maximum luminosity to within $\sim$ 10% and since the maximum luminosity is the same within $\sim$ 15% for different globular cluster sources, the "true'' critical luminosity can only be regarded as an approximate empirical standard candle.

In Fig. 7 we show in grey bands the range in the Eddington luminosity limit for matter with cosmic composition (X=0.73; lower band) and hydrogen-poor matter (X=0; upper band) for a (canonical) 1.4 $M_{\odot}$ neutron star with a radius of 10 km. The upper and lower boundaries of each band is determined by the Eddington limit observed by an observer far away from the neutron star with very strong photospheric radius expansion and very weak photospheric radius expansion, respectively (van Paradijs 1979; Goldman 1979; see Lewin et al. 1993). Note that an increase in the neutron star mass shifts the band slightly upwards as well as substantially broadens it. A higher than canonical neutron star mass seems to be required in some models explaining the kHz QPO in neutron star low-mass X-ray binaries (see e.g. Stella et al. 1999; Stella 2001; Lamb & Miller 2001), and is derived dynamically for the low-mass X-ray binary and X-ray burster Cyg X-2 (Orosz & Kuulkers 1999). Our critical luminosity is close to the Eddington luminosity limit for hydrogen-poor matter. Therefore, our critical luminosity is most likely this Eddington limit (see also e.g. van Paradijs 1978).

Previously, "super-Eddington luminosities'' were reported for several sources in our sample (e.g. Grindlay et al. 1980; van Paradijs 1981; Inoue et al. 1981; see also Verbunt et al. 1984) and it was referred to as the "super Eddington limit problem'' (e.g. Lewin 1985). Winds or outflows from the neutron-star surface set-up by the high radiation pressure were introduced to resolve (part of) this problem (e.g. Melia & Joss 1985; Stollman & van Paradijs 1985). Our findings do not support super-Eddington luminosities, except possibly for 4U 2129+12 (note that 4U 2129+12 is consistent with the Eddington limit for hydrogen-poor matter if we assume e.g. a somewhat higher neutron star mass, see above). We attribute this mainly to a better knowledge of the distance and their uncertainties.

Five bursters (including GRS 1747-312) in our sample show bolometric peak X-ray burst luminosities near $1.5 \times 10 ^{38}$ erg s^-1, which is comparable to the Eddington limit of matter with solar composition. Of these, only the sources 4U 1746-37 and GRS 1747-312 showed X-ray bursts with evidence for photospheric radius expansion; the maximum peak X-ray burst luminosity for the other sources could have been higher if radius expansion bursts were seen.

If there is a bimodal distribution of the bolometric X-ray burst peak fluxes it is consistent with the thought that the donors in systems with an ultra-short orbital period ( $\mbox{$\la$ }$ 1 hr: MX 0513-40, H1825-331, 4U 1820-30, A1850-08; see Table 2) do not provide hydrogen, whereas the donors in systems with a longer orbital period (4U 1746-37, GRS 1747-312; see Table 2 ) do provide hydrogen-rich material (e.g. Nelson et al. 1986; Verbunt 1987; see also Juett et al. 2001). The short duration of the X-ray bursts from XB 1745-25 and 4U 1820-30 ( $\mbox{$\la$ }$ 25 s) is consistent with the fuel being only helium (Bildsten 1995). However, the X-ray bursts with longer duration ( $\sim$ mins) from the other systems can not be accomodated this way (e.g. Fujimoto et al. 1981; Bildsten 1998, and references therein), unless the mass accretion rate onto the neutron star is rather low (see Bildsten 1995). On the other hand, we note that the Eddington limit of hydrogen poor matter may still be reached during long X-ray bursts if a hydrogen rich envelope is ejected during the photospheric radius expansion stage of a helium flash (Sugimoto et al. 1984).

5 Discussion